40 2. Background it is negative another equation for the same plane may be obtained by multiplying through by −1. It is not without loss of generality to assume the coeﬃcient of x is 1, since the plane defined by y + z = 0, for example, has no equation with an x-coeﬃcient of 1. Exercise 2.6.1. Here are some proofs you can try that don’t involve induction: (i) ¬(∀m)(∀n)(3m + 5n = 12) (over N) (ii) For any integer n, the number n2 + n + 1 is odd. (iii) If every even natural number greater than 2 is the sum of two primes, then every odd natural number greater than 5 is the sum of three primes. (iv) For nonempty sets A and B, A × B = B × A if and only if A = B. (v) √ 2 is irrational. (Hint: work by contradiction, assuming there is a fraction of integers in least terms that equals √ 2.)

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